Graphical Models for Time-Series

D

igital Object Identifier 10.1109/MSP.2010.938028? PHOTODISC [ D avid Barber and A. Taylan Cemgil ]

T

ime-series analysis is cen-tral to many problems in signal processing, including acoustics, image processing, vision, tracking, informa-tion retrieval, and finance, to name a few [1], [2]. Because of the wide base of application areas, having a common description of the models is useful in trans

f errin

g ideas between the various communities. Graphical models provide a compact way to represent suc

h models and thereby rapidly transfer ideas. We will discuss briefly how classical time-series models such as Kalman filters and hidden Markov models (HMMs) can be represented as graphical models and critically how this representation differs from other common graphical representations such as state-transition and block diagrams. We will use this framework to show how one may easily envis-age novel models and gain insight into their computational implementation.

TIME-SERIES AND GRAPHICAL MODELS

Classically, time-series analysis falls into two camps in which the central assumption is that the process generating the time series is continuous or alternatively discrete. In the continuous case, classical textbook methods such as Kalman filters depend

heavily on linear dynamical systems (LDSs) for which the underlying theory is well understood [3]. In the discrete case, the well-known HMM has enjoyed considerable success [4]. However, recent developments in engineering, sta-tistics, and machine learning consider underlying processes that can be both discrete and continuous. Such models are natural in many applications in con-trol, tracking, and signal processing where one may wish to discover step-changes in an underlying continuous

dynamical process, such as might occur

for example with a fault. Working with these increasingly sophisticated models requires specialized treatments and often approximations [5]. There are, however, important special cases, such as the reset models, where the computational complexity of inference is r elatively modest [6]. Here we take advantage of the graphical models framework to describe some of the basic time-series models, their extensions, and applications in signal processing.

DEVELOPING A GRAPHICAL REPRESENTATION

A probabilistic model of a time series y 1:T 55y 1,c ,y T 6 is a specification of a joint distribution p 1y 1:T 2. In time series, it is nat-ural to consider models consistent with the causal nature of time. To achieve this, we may use Bayes’ rule of the probability of A con-ditioned on knowing

B ,p 1A |B 25p 1A ,B 2/p 1B 2, and write

[Gaining insight into

their computational implementation ]

where pa 1y i 2 denotes the set of parental variables for variable y i . We depict a belief network using a graph in which a node represents a variable y i and the variables that point to y i are the parents of this variable. Each node in the belief net-work then corresponds to a factor in the joint distribution over all variables; see Figure 1. By Bayes’ recursive construction, the graph must be acyclic. The most general form of belief network is therefore the cascade graph in which the parents of a variable are all the previous variables in the ordering. Any valid belief network can be obtained by removing edges in the cascade graph, with each removal corresponding to a condi-tional independence assumption. The first-order Markov model can be represented in this form in which i indexes time and pa 1y i 25y i 21. A second-order Markov model has pa 1

y i 255y i 21, y i 226. As an example, the classical L th-order

scalar auto-regressive (AR) model y t 5g L

l 51a l y t 2l 1P t for c o e f f i c i e n t s a l , l 51,c , L a n d G a u s s i a n n o i s e P t ,

N 1P t | 0, s 22 corresponds to the transition

p 1y t |

y t 2L : t 2125N a y t a L

l 51

a l y t 2l , s 2b

with a belief network representation in which the parent set of each variable contains the previous L observations. When the parameters of the model are also unknown, they can be incorpo-rated into the graphical description as well; see “Parameter Learning.” Graphs have a long history in the description of

[FIG1] Belief network representations of time-series models: (a) Cascade graph. (b) First-order Markov model

p ( y 4|y 3)

p (

y 3|y 2) p ( y 2|y 1) p (

y 1). (c) Second-order Markov model p (

y 4|y 3, y 2) p ( y 3|y 2, y 1) p ( y 2|y 1) p (

y 1).

t ime-series models and it’s important to stress the difference between a probabilistic graphical model and alternative graph rep-resentations such as state transi-tion diagrams or block diagrams that use an entirely different set of semantic rules.More generally, probabilistic graphical models are compact depictions of independence and factorization assumptions of a probability density. Besides the directed acyclic graphs, there exist also other formalisms. Two well-known formalisms are undirected models [7] and factor graphs [8]. To keep this sur-vey self-contained, we focus only on the case of directed graphical https://www.wendangku.net/doc/bc9107962.html,TENT MARKOV MODELS A more general framework for modeling time-series data uses a latent, unobserved variable x t , from which the observations y t are generated [9]. For exam-ple, in tracking, x t might rep-resent the position of an object

that is assumed to move according a transition dynam-ics p 1x t | x t 212. However, we

cannot directly observe x t , but some noisy function of it p 1y t |x t 2—for example a noisy radar reading y t of the approxi-mate distance to the object. We would like to use the observa-tions y 1,c , y t to track the likely position x t of the object. Due to their development in different research communities, latent Markov models are variously called state-space models

or HMMs. We use the term HMM to refer to a latent Markov model with discrete latent states. Both the classical Kalman

filter and the HMMs have the same belief network

r epresentation, (see Figure 2), differing only in the domain of the variables and the specifics of the transition and observa-tion model.

DISCRETE LATENT STATE MARKOV MODELS

HMMs are models in which the latent variables x t are discrete [4]. The observations y t can be discrete or continuous. Since the latent x t are discrete, HMMs are able to model discrete changes in the underlying state. To emphasize that the x t are discrete, graphically, we use a square node. For example, in a switching AR (SAR) model, a set of S different AR models is available, and x t [51,c , S 6 may be used to indicate which of the AR models is to be used at time t ; see Figure 3. In Figure 4, a segment of a speech signal is shown; each of the ten available AR models is responsible for modeling the dynamics of a basic subunit of speech [10], [11]. The interest is to determine when each subunit is most likely to be active. This corresponds to the computation of the most-likely path x 1:T given the observed signal p 1x 1:T |y 1:T 2. Typically we use a lower-case version of a variable to denote instan-tiation. While models such as the SAR model contain both discrete and continuous variables, fundamentally, the underlying latent process is discrete.

CONTINUOUS STATE LATENT MARKOV MODELS

Dealing with continuous variable distributions is generally awkward and the set of models that are analytically tractable is limited. Within this tractable class, the LDS plays a special role, being essentially the continuous analog of the discrete-state HMM. An LDS has the following form [2], [3]:

x t 5Ax t 211P t , y t 5Cx t 1n t ,

where the noise terms P t and n t are Gaussian distributed. This is more commonly referred to as a Kalman filter in the signal processing literature. The traditional focus is the use of this linear system to compute quantities of interest, in particular the expected mean of x t given past observations. This termi-nology unfortunately confuses the distinction between a model

[FIG3] A switching (second order) AR model. Here the x t

indicates which of a set of available AR models is active at time t . In terms of inference, conditioned on y 1:T (this is an HMM).

[FIG4] A spoken digit of the word “four” modeled by a SAR

model. The SAR model was trained on many example sequences using S 510 states with a left-to-right transition matrix. Given the particular audio sequence shown, the most likely set of states x 1:T are computed. The colors indicate the states used at each time. The states found correspond to basic sound component models that when used in sequence generate realistic sounding waveforms.

[FIG2] A first-order state-space model with “hidden” variables. For discrete hidden variables x t [51,c , H 6, t 51:T the model is termed an HMM.

PROBABILISTIC GRAPHICAL MODELS ARE COMPACT DEPICTIONS OF INDEPENDENCE AND FACTORIZATION ASSUMPTIONS OF A PROBABILITY DENSITY.

and the use of it to infer a quantity of interest. As such most students are familiar with the Kalman filter from an

a lgorithmic perspective, unaware that the algorithm is an instance of the generic filtering algorithm for all graphs con-sistent with the belief network Figure 2. As a probabilistic model, the LDS corresponds to

p 1x t | x t 2125N 1x t | Ax t 21, Q 2, p 1y t | x t 25N 1y t | Cx t , R 2.The model is completed by choosing a suitable prior: p 1x 125N 1x 1 | m 1, P 12. Furthermore, as we describe below, infer-ence in this model is computationally straightforward. The graphi-cal models perspective emphasizes the application of the independence assumptions of the model to derive generic recur-sions; these recursions may be then implemented in specific numerical instances of distributions consistent with the graphical model representation.

INFERENCE IN LATENT MARKOV MODELS

Latent Markov models have widespread application in a variety of tracking domains, for which one often wishes to infer the distribution of the latent state x t based on noisy observations. These will be derived for general latent variables x t using the notation e dX that either integrates or sums over the domain of X . The important conclusion we shall reach is that the same procedure applies in all models consistent with the belief net-work Figure 2, irrespective of the numerical form of the tran-sition and observation distributions. As we also discuss, while the general procedure produces exact results, it can only be numerically implemented in a restricted class of transition and observation distributions, the two most common being i) dis-crete latent variables (HMM) and ii) linear Gaussian transition and observations (LDS), giving rise to the classical Kalman fil-ter. We sketch below the generic filtering and smoothing recursions for the class of all latent Markov models.

FILTERING: COMPUTING p(x t | y 1:t )

Filtering is the estimation of the current state given the observations so far. It is useful to first compute the joint marginal p 1x t , y 1:t 2 since the likelihood of the sequence can be obtained from this expression. A recursion for p 1x t , y 1:t 2 is obtained by considering the conditional independence assumptions of the model

p 1x t , y 1:t 253dx t 21p 1y t | x t 2p 1x t | x t 212p 1x t 21, y 1:t 212. (4)

Hence, if we define a 1x t 25p 1x t , y 1:t 2 with a 1x 125p 1y 1|x 12p 1x 12 we have the so-called a -recursion

a 1x t 25p 1y t | x t 2 3dx t 21p 1x t | x t 212a 1x t 212, t .1. (5)

This recursion has the interpretation that the filtered distribution a 1x t 212 is propagated forwards by the dynamics for one time step

to reveal a new “prior” distribution at time t . Normalization gives the filtered posterior p 1x t | y 1:t 2~a 1x t 2.

PARALLEL (FORWARD-BACKWARD) SMOOTHER p (x t | y 1:t )

In parallel smoothing, one separates the smoothed posterior into contributions from the past and future

p 1x t , y 1:T 25p 1x t , y 1:t , y t 11:T 25p 1x t , y 1:t 2 p 1y t 11:T | x t , y 1:t 2

5a 1x t 2b 1x t 2. (6)

The term a 1x t 2 is obtained from the “forward” a recursion (5). The term b 1x t 2 may be obtained using a “backward” b recursion with b 1x T 251

b 1x t 21253dx t p 1y t | x t 2p 1x t | x t 212b 1x t 2, 2#t #T .

SEQUENTIAL (CORRECTION) SMOOTHER p(x t | y 1:T )

The parallel smoothing method given above is perhaps best known in the HMM literature [4]. Particularly in the case of continuous variables, however, some care is required with its numerical imple-mentation [12]. In practice, it is often more suitable to use a sequential method that is based on the fact that conditioning on the present makes the future redundant [13] p 1x t | y 1:T 253dx t 11p 1x t , x t 11 | y 1:T 2

53dx t 11p 1x t | x t 11, y 1:t , y t 11:T 2p 1x t 11 | y 1:T 2. (7)

This then gives a recursion for g 1x t 2;p 1x t | y 1:T 2

g 1x t 253dx t 11p 1x t | x t 11, y 1:t 2g 1x t 112 (8)

with g 1x T 2~a 1x T 2. The term p 1x t | x t 11, y 1:t 2 may be computed based on the filtered results p 1x t | y 1:t 2 using a dynamics reversal step

p 1x t | x t 11, y 1:t 2~p 1x t 11, x t | y 1:t 25p 1x t 11 | x t 2p 1x t | y 1:t 2, (9)where the proportionality constant is found by normalization. The procedure is sequential since we need to first complete the a recursions, after which the g recursion may begin. This is also called a correction smoother, since it takes the filtered results and corrects them into smoothed results. A significant advantage of this sequential approach is that the recursion deals directly with densities, unlike the parallel approach that forms a recursion for a quantity that is itself not a density in x t . This has important bene-fits for models (such as the SLDS described below) for which exact smoothing is not computationally feasible.

INFERENCE IN LINEAR DYNAMICAL SYSTEMS

Filtering and smoothing for the LDS follows the general approach, with the most common smoothing approach being the

x

predictor

s

corrector

u

future

s

past

sequential method. Since all

updates for the LDS are linear-Gaussian, the filtered and

smoothed distributions are

Gaussians. The a and g recur-sions can thus be represented by

updates to the mean and covari-ance of the distributions. Working out these updates is a stan-dard exercise in multivariate Gaussian integration resulting in

the well-known Kalman filtering and smoothing recursions [2].

THE SWITCHING LINEAR DYNAMIC SYSTEM The HMM and LDS are two classical signal processing models. A more complex model is the switching LDS (SLDS) that mar-ries the HMM and LDS by breaking the time series into s egments, each modeled by a potentially different LDS; see Figure 5. Such models can handle situations in which the underlying linear model jumps from one parameter setting to another. Thus, the latent process contains both discrete and

continuous variables. The SLDS is an attractive model and used in many d isciplines, from econometrics to machine learning [14]–[17]. At each time t , a switch variable s t [1,c , S selects a single LDS from the available set. The dynamics of s t itself is Markovian, with transition p 1s t | s t 212. The probabilistic model defines a joint distribution p 1y 1:T , x 1:T , s 1:T 25q T t 51

p 1y t | x t , s t 2p 1x t | x t 21, s t 2p 1s t | s t 212with

p 1y t | x t , s t 25N 1y t | C 1s t 2x t , R 1s t 22,

p 1x t | x t 21, s t 25N 1x t | A 1s t 2x t 21, Q 1s t 22.At time t 51, p 1s 1 | x 1, s 12 denotes the prior p 1s 12, and

p 1x 1 | x 1, s 12 denotes p 1x 1 | s 12. Due to its popularity in many

different fields the SLDS has many different names; it is also

called a jump Markov model/process, switching Kalman filter,

switching linear Gaussian state-space model, conditional linear Gaussian model. Given its importance, we will spend some time considering the particular issues in dealing with the SLDS.EXACT INFERENCE IS COMPUTATIONALLY INTRACTABLE In terms of the cluster variables z 1:T , with z t ;1s t , x t 2 and visible variables y 1:T , the belief network of the SLDS is a latent Markov

model, for which the exact filtering and smoothing recursions are given in the section “Inference in Latent Markov Models.” One might therefore envisage no difficulty in carrying out inference. However, both exact filtered and smoothed inference in the SLDS

is intractable, s caling exponentially with time. As an informal explanation, consider filtered posterior inference, for which the forward pass is, by analogy with (5), a 1s t , x t 25p 1y t | x t , s t 2

3a s t 213x t 21

p 1s t , x t | s t 21, x t 21, y t 2a 1s t 21, x t 212. (10)

At time step 1, a 1s 1, x 12~p 1x 1 | s 1, y 12 p 1s 1 | y 12 is an indexed set of Gaussians. At time step 2, due to the summation over

the states s 1, a 1s 2, x 22 will be an indexed set of S Gaussians;

similarly at time step 3, it will be S 2 and, in general, gives

rise to exponentially many Gaussians, S t 21, at time t . The origin of the intractability of the SLDS therefore differs from

structural intractability, resulting from the inability to form a singly connected structure by the clustering of a small

number of variables [18]. Since filtering and smoothing in

the SLDS require some form of approximation, we therefore

have to choose which approximation strategy to follow. Approximate inference in the SLDS has a large associated lit-erature describing available techniques that range from Monte Carlo methods to deterministic variational techniques [19], [20], [15]. One of the most robust techniques is the Gaussian sum approximation and, rather than giving a s urvey on the a vailable techniques, we outline the rationale for this method below.GAUSSIAN SUM FILTERING

A popular approximate SLDS filtering scheme is to keep in check the exponential explosion in the number of Gaussian components by projecting each filtered update to a limited number of components. A graphical depiction is given in Figure 6. At each stage, a single Gaussian component is propa-gated forwards by the S separate LDS dynamics, each giving rise to a s eparate filtered distribution according to LDS filter-ing. Sub s equently, this S 2 Gaussian mixture is collapsed back to an S component Gaussian mixture, preventing the exponen-tial explosion in mixture components. Such Gaussian sum

[FIG5] The independence structure of the SLDS. Square nodes s t

denote discrete switch variables; x t are continuous latent/hidden variables and y t continuous observed/visible variables. The discrete state s t determines which LDS from a finite set of LDSs is operational at time t .

MOST STUDENTS ARE FAMILIAR

WITH THE KALMAN FILTER, UNAWARE THAT THE ALGORITHM IS AN INSTANCE OF THE GENERIC FILTERING ALGORITHM.

f ilterin

g approximations were developed early in the literature, in particular in the work by Alspac

h and Sorenson [21]. The method is a form of the general approximation class called assumed density filtering in which an approximate mixture density is projected back to a chosen approximation family at each update [22]. The complexity of the resulting approximate forward pass is O1ISTL2, where I is the number of mixture components of the collapsed distribution and L is the cost of performing a filtered update for the LDS. The recursion is ini-tialized with p1x1, s1 | y12~p1y1 | x1, s12p1x1 | s12p1s12, where p1x1|s12 and p1s12 are given prior distributions. GAUSSIAN SUM SMOOTHING

The g recursion (8) suggests a convenient Gaussian sum smoothing approximation. Since the g recursion can be interpreted as a backwards dynamics, one may propagate each component in a Gaussian sum smoothed approxima-tion backwards according to each of the S dynamical sys-tems. This results in an S2 component Gaussian mixture distribution which, analogously to filtering, may be col-lapsed back to a smaller number of components to prevent the exponential explosion in components. A popular stan-dard method to achieve this is called generalized pseudo-Bayes [15]. An alternative approach, which makes less severe approximation assumptions, is expectation-correction [23], which we use throughout our examples.

NOISY SIGNAL RECONSTRUCTION

Continuous observation models such as the SAR model have been successfully applied in many areas of signal processing, including audio signal processing [10]. Since such models are essentially HMMs, however, they are not well suited to signal reconstruction in which we aim to infer a clean continuous

signal from a noisy observation. A natural extension is to include additional graphical links from the AR output y t to form a noisy observation y|t. For example, additive zero mean Gaussian noise with variance s y2 can be expressed as p1y|t| y t25N1y|t|y t, s y22. Given the noisy observation seq-uence y|1:T, our interest is then to reconstruct a clean signal y1:T, based on the assumption that the clean signal is itself expressed as a SAR model; see Figure 7. This model is a form of SLDS and may be used to form noise-robust speech recog-nition systems [11]; see Figure 8.

[FIG6] Gaussian sum filtering. (a) Depicts the previous Gaussian mixture approximation a1x t, s t2 for two states S52 (red and blue) and I53 mixture components. The area of each ellipse corresponds here to the relative weight of each component rather than the variance. There are S52 different linear systems that take each of the components of the mixture into a new filtered state, the color of the arrow indicating which dynamic system is used. After one time step, each mixture component branches into a further S components so that the joint approximation a1x t11, s t112 contains (b) S2I components. T o keep the representation computationally tractable, the mixture of Gaussians for each state s t11 is collapsed back to I components. This means that each colored state needs to be approximated by a smaller I component mixture of Gaussians. There are many ways to achieve this. A na?ve but computationally efficient approach is to simply ignore the lowest weight components, as depicted in (c).

TRAFFIC FLOW As an illustration of modeling and inference with a SLDS consider a traffic network; see Figure 9. There are four junctions a , b , c , d and traffic flows along the roads in the direction indicated. Traffic flows into junction a and then goes via different routes to d . Flow out of a junction must match the flow in to a junction (up to noise). There are traffic light switches at junctions a and b that, depending on their state, route traffic differently along the roads. Then using f to denote flow, we model the flows using the switching linear system

f a 1t 2f a S d 1t 2f a S b 1t 2f b S d 1t 2f b S c 1t 2f c S d 1t 2v 5f f a 1t 212

f a 1t 2

1210.7533s a 1t 25141133s a 1t 25242

f a 1t 21210.2533s a 1t 25141133s a 1t 25342f a S b 1t 21210.533s b 1

t 25142f a S b 1t 21210.533s b 1t 25141133s b 1t 25242f b S c 1t 212.

In the above, 3A 451 if A is true

and is zero otherwise. By identify-ing the flows at time t with a six-dimensional vector hidden variable x t , we can write the

above flow equations as an SLDS

in x t for a set of suitably defined matrices A 1s 2 where the switch variable s 5s a z s b , takes 33256 states. The switch variables follow a simple Markov transition p 1s t | s t 212 that biases the switches to remain in the same state in preference to jumping to another state. We addi-tionally include small noise terms to model cars parking or deparking during a single time frame. The noise is larger at the inflow point a to model that the total volume of traffic

entering the system can vary. Noisy measurements of the flow

into the network are taken at a and d . Given an observed

sequence 1a t , d t 2, t 51,c , 100 (see Figure 10), the task is to infer the filtered and smoothed traffic flows throughout the network. A na?ve approximation based on discretizing each continuous flow into 20 bins would contain 2333206 or

[FIG9] A representation of the traffic flow between junctions at

a ,

b ,

c ,

d , with traffic lights at a and b . If s a 51, a S d and a S b carry 0.75 and 0.25 of th

e flow out o

f a , respectively. If s a 52, all the flow from a goes through a S d ; for s a 53, all the flow goes through a S b . For s b 51, the flow out of b is split equally between b S d and b S c . For s b 52, all flow out of b goes alon

g b S c .

[FIG10] Time evolution of the traffic flow measured at two

points in the network. (a) Sensors measure the total flow into the network f a 1t 2 and (b) the total flow out of the network, f d 1t 25f a S d 1t 21f b S d 1t 21f c S d 1t 2 . The total inflow at a undergoes a random walk. Note that the flow measured at d can momentarily drop to zero if all traffic is routed through a S b S c in two consecutive time steps.

RESET MODELS ARE SPECIAL SWITCHING MODELS IN WHICH THE SWITCH CAN RESET THE LATENT STATE, ISOLATING THE PRESENT FROM THE PAST.[FIG7] A latent switching (second order) AR model. Here the x t indicates which of a set of available AR models is active at time t . The “clean” AR signal y t , which is not observed, is corrupted

by additive noise to form the noisy observations y |t

. In terms of inference, conditioned on y |1:T , this can be expressed as a SLDS.

[FIG8] Signal reconstruction using a latent left-right SAR model; see Figure 7. (a) Noisy signal y |1:T . (b) Reconstructed clean signal y 1:T . The dashed lines and the numbers show the most-likely state segmentation s 1:T *.

384 million states. Even for such a modest-size problem, an approximation based on a simple discretization is therefore impractical. As a practical alternative, filtering and smoothing for this SLDS can be carried out using a Gaussian sum approx-imation; see Figure 11.

RESET MODELS

While switching models such as the SLDS are powerful, they are computationally difficult to implement. As such, it is interesting to consider special cases for which inference is computationally sim-pler. Reset models are special switching models in which the switch can reset the latent x t , isolating the present from the past. These models are also known as change-point models [6], though the term is less precisely defined. We use the state c t 51, to denote a “change” that resets x t , independent of the past, and c t 50 to denote that the standard dynamics continues. Then

p 1x t | x t 21, c t 25e p 01x t | x t 212c t 50

p 1

1x t 2

c t 51

.Similarly, we write

p 1y t | x t , c t 25e p 01y t | x t 2

c t 50

p 1

1y t | x t 2

c t 51

.The switch dynamics are first-order Markov with transition p 1c t | c t 212. Under this model the dynamics follows a standard

system p 01x t | x t 212 until c t 51 when the continuous state is drawn from a “reset” distribution p 11x t 2, independent of the past; see Figure 12. Such models are of interest when the time series is following a trend but suddenly changes and the past is forgotten. An SLDS with S 52 states, one of which resets the continuous dynamics, is an example of such a reset model. Importantly, the complexity of filtered inference scales with O 1LT 22, compared to O 1LT 2T 2 in the general two-state switch-ing case, as discussed in “Filtering in the Reset Model.”

[FIG11] Given the observations from Figure 10, we infer the flows and switch states of all the latent variables. (a) The correct latent flows through time along with the switch variable state used to generate the data. The colors correspond to the colored edges and nodes in Figure 9. (b) Filtered flows based on a I 52 Gaussian sum forward pass approximation. The filtered traffic light states of s a and s b are plotted below. (c) Smoothed flows and corresponding smoothed traffic light states using a two-component Gaussian sum smoothing approximation.

020406080100020406080100020406080100

20406080100204060801002040608010020

40

60

80

100

20

40

60

80

100

[FIG12] The independence structure of a reset model. Square nodes c t denote the binary reset variables and s t the state dynamics. The x t are continuous variables, and y t continuous observations. If the dynamics resets, the dependence of the continuous x t on the past is cut.

POISSON RESET MODEL

Reset models are not limited to conditionally Gaussian cases. To illustrate this, we consider the following Poisson model. At each time t , we observe a count y t that we assume is Poisson distributed with an unknown positive intensity x t . The intensity is constant, but at certain unknown times t , it jumps to a new value. The indicator variable c t denotes whether time t is such a change point or not. Mathematically, the model is

p 1x t |x t 21, c t 253c t 504 d 1x t 2x t 21213c t 514 G 1x t ; n , b 2, t $2 (14)

p 1y t | x t 25P O 1y t ; x t 2, p 1c t 25B E 1c t ; p 2 (15)

with p 1x 125G 1x 1; a 1, b 12. The symbols G , BE , and P O denote the Gamma, Bernoulli, and the Poisson densities, resp e ctively. Given observed counts y 1:T , the task is to find the posterior probability of a change and the associated inten s ity levels for each region between two consecutive change points. Plugging the above definitions in the generic updates (11) and (12), we see that a 1x t , c t 502 is a Gamma potential, and that a 1x t , c t 512 is a mixture of Gamma potentials, where a Gamma potential is defined as

f 1x 25e l G 1x ; a , b 2 (16)

via the triple 1a , b , l 2. For the corrector update step, we need to calculate the product of a Poisson term with the observation model p 1y t | x t 25P O 1y t ; x t 2. A useful proper-ty of the Poisson distribution is that, given the observa-tion, the latent variable is Gamma distributed as

P O 1y ; x 25G 1x ; y 11, 12. Hence, the update equation requires multiplication of two Gamma potentials. A nice property of the Gamma density is that the product of two Gamma densities is another Gamma potential. The a recur-sions for this reset model are therefore closed in the space of a mixture of Gamma potentials, with an additional Gamma potential in the mixture at each time step. A similar approach can be used to form the smoothing recursions.We illustrate the algorithm on a coal mining disaster data set [24]. The data consists of the number of deadly coal-mining disasters in England per year over a time span of 112 years from 1851 to 1962. It is widely agreed in the statistical literature that a change in the intensity (the expected value of the number of disasters) occurs around the year 1890, after new health and safety regulations were introduced. In Figure 13, we show the marginals p 1x t | y 1:T 2 along with the filtering density. Note that we are not constraining the number of change points a priori and in principle allow any number. The smoothed densities indeed suggest a sharp decrease around t 51890.

RESET HIDDEN MARKOV MODEL

The reset models described are useful in many applications but limited since only a single standard dynamics is avail-able. An important extension is to consider a set of avail-able dynamical models, indexed by s t [51,c , S 6, with a reset that cuts dependency of the continuous variable on the past [17]

p 1x t | x t 21, s t , c t 25e

p 01x t | x t 21, s t 2c t 50

p 11x t | s t 2

c t 51

. (17)The states s t follow a Markovian dynamics p 1s t | s t 21, c t 212; see Figure 14. A reset occurs if the state s t changes, otherwise, no reset occurs

p 1c t 51| s t , s t 21253s t 2s t 214. (18)The computational complexity of fil-tering for this model is O 1LS 2T 22 that can be understood by analogy with the reset a recursions, (11), (12) on replac-ing x t by 1x t , s t 2. In the next section, we describe a signal processing appli-cation for this model.

DYNAMIC HARMONIC

MODEL AND RESET MODELS

A key problem in music signal process-ing is music transcription; the identifi-cation of note events. The fundamental frequency, corresponding to the largest common divisor of mode frequencies, is perceived as the pitch or “note” in

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F i l t e r e d I n t e n s i t y 2

46S m o o t h e d I n t e n s i t y

Y ear 2

46(a)

(b)(c)

[FIG13] Estimation of change points on the coal mining disaster data set. (a) The number of deadly disasters each year. (b) Filtered estimate of the marginal intensity p 1x t

|

y 1:t 2. Here, darker color means higher probability. (c) Smoothed estimate p 1x t |

y 1:T 2.

music. For transcription, we estimate which note is played and when. The polyphonic case assumes that more than one note may be played at any time. Here we concentrate on the monophonic case of a single note s t [51,c , S 6 being played at any time. For each note s t , a musical instrument creates oscillations with

modes at frequencies that are roughly related by ratios of

integers. One can model this using an LDS that consists of

a bank of “phasors”

A 1s t 25diag 1Z 11s t 2,c , Z n 1s t 2,c , Z W 1s t 22, (19)where each phasor corresponds to a rotation matrix around a

multiple n of a fundamental frequency v 1s t 2

Z n 1s t 25e 2ng 1s t 2a

cos 1nv 1s t 22

sin 1nv 1s t 22

2sin 1nv 1s t 22

cos 1nv 1s t 22

b .

(20)

Here, the index n gives the number of the harmonic and sets both

the damping parameter g and the frequency v . The dynamics of a single phasor is plotted in Figure 15. A note changes when s t 2s t 21 at which point we assume the continuous dynamics is reset. Assuming a simple Markov model dynamics for the notes s t , the model is p 1x t | x t 21, s t , c t 253c t 504N 1x t | A 1s t 2x t 21, qI 2

13c t 514 N 1x t | 0, QI 2 (21)

p 1y t |x t 25N 1y t | Cx t , r 2, (22)

where C 531 0 1 c 1 04 is a projection matrix that sums the first components of each phasor and the observation

noise variance is r . The identity

matrix is denoted by I and q

and Q are transition variances with Q W q . This model is then

a reset HMM model, for which

the computations can be car-ried out efficiently.An example from transcribing a real guitar recording is

presented in Figure 16. An interesting, yet more complex problem is to deal with polyphony, or “chords.” That is when more than one note can be simultaneously present. Using the graphical models perspective, this is straightforward to achieve by using a factorial construction in which each constituent of

the chord is modeled with an independent reset HMM model.

The elements of the chord are coupled via an observation model that combines all elements into a scalar observation at each time [25], [17]. DISCUSSION

We presented an overview of the graphical models viewpoint of time-series modeling. Graphical models provide a compact description of the basic independence assumptions behind a model and, as such, are a useful way to communicate ideas. This also stresses general-purpose inference routines, for which classical algorithms such as the Kalman filter or for-ward-backward in the HMM are special cases.

Using graphical models, it is easy to envisage new models tailored for a particular environment. For example, we

h ighlighted the switching state-space models and their [FIG14] The independence structure of a reset HMM model. Square nodes c t denote binary change point variables, x t are continuous latent variables, and y t continuous observations. The discrete state s t determines which LDS from a finite set of LDSs is operational at time t .

[FIG15] A single phasor plotted as a damped two dimensional rotation. By taking a projection onto the y axis, the phasor generates a damped sinusoid.

[FIG16] Note transcription of a signal recorded from a bass

guitar playing a major scale. (a) Raw acoustic signal. (b) The most probable joint note trajectory is shown with the vertical axis denoting the note index.

USING GRAPHICAL MODELS, IT IS

EASY TO ENVISAGE NEW MODELS TAILORED FOR A PARTICULAR ENVIRONMENT.

p otential in signal processing applications. Such models are natural extensions of traditional signal p rocessing techniques that are limited to the assumption that the underlying pro-cess generating the data is either discrete or continuous. In particular, the SLDS is a simple graphical marriage of a con-tinuous and discrete latent Markov model. We showed how these models can be used to detect changes in the underlying dynamics of a system, and gave examples of their use in sig-nal reconstruction and system monitoring.

Issues with computational tractability do not magically disappear within this framework, and the switching models are f ormally computationally intractable in general. Nevertheless, in some cases simple deterministic approxima-tions based on mixture models can be effective, for which the graphical model helps guide intuition in the approximation. Alternatives to the deterministic approximation method we discussed are based on Monte Carlo sampling. Typical strate-gies use Markov chain Monte Carlo (MCMC)or sequential Monte Carlo, also known as sequential importance sampling or particle filtering [26], [27], with specialized algorithms designed for switching state-space models; for MCMC (see [28] and [29]).

The effective application of switching models in the real world is gaining pace, partly through the restricted reset mod-els, but also via increased computational power that brings the more general models into consideration through carefully developed approximations. As such, developing new models and associated approximate inference schemes is likely to remain an active area of research, with graphical models play-ing an important role in facilitating communication and guid-ing intuition.

AUTHORS

David Barber (D.Barber@https://www.wendangku.net/doc/bc9107962.html,) received his B.A. degree in mathematics from Cambridge University and his Ph.D. degree in theoretical physics (statistical mechanics) from Edinburgh University. He is currently a reader in information processing in the Department of Computer Science, University College London (UCL), where he develops novel information processing schemes, mainly based on the application of proba-bilistic reasoning. Prior to joining UCL he was a lecturer at Aston and Edinburgh Universities.

A. Taylan Cemgil (taylan.cemgil@https://www.wendangku.net/doc/bc9107962.html,.tr) received his

B.Sc. and M.Sc. degrees in computer engineering from Bogazici University, Turkey. He received his Ph.D. degree from Radboud University, Nijmegen, The Netherlands, in 2004. He worked as a postdoctoral researcher at the University of Amsterdam and the University of Cambridge. Since 2008, he has been an assistant professor of computer engineering at Bogazici University. His research is focused on developing computational techniques for statistical information process-ing. He is a Member of the IEEE. REFERENCES

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详解特斯拉Model S 1、Model S的核心技术是什么？ 核心技术是软件，主要包括电池管理软件，电机以及车载设备的电控技术。最重要的是电池控制技术。 Model S的加速性能，续航里程、操控性能的基础都是电池控制技术，没有电池控制技术，一切都就没有了。 2、Model S的电池控制技术有什么特色？ 顶配的Model S使用了接近7000块松下NCR 18650 3100mah电池，对电池两次分组，做串并联。设置传感器，感知每块电池的工作状态和温度情况，由电池控制系统进行控制。防止出现过热短路温度差异等危险情况。 在日常使用中，保证电池在大电流冲放电下的安全性。 其他厂商都采用大电池，最多只有几百块，也没有精确到每块电池的控制系统。 3、为什么要搞这么复杂的电池控制系统？ 为了能够使用高性能的18650钴锂电池。高性能电池带来高性能车。因为18650钴锂电池的高危性，没有一套靠谱的系统，安全性就不能保证。这也是大多数厂商无论电力车，插电车，混合动力车都不太敢用钴锂电池，特别是大容量钴锂电池的原因。 松下NCR 18650 3100mah，除了测试一致性最好，充放电次数多，安全性相对较好以外，最重要的是能量大，重量轻，价格也不高。 由于能量大，重量轻，在轿车2吨以内的车重限制下，可以塞进去更多的电池，从而保证更长的续航里程。因为电池输出电流有限制，电池越多，输出电流越大，功率越大，可以使用的电机功率也就越大。电机功率越大，相当于发动机功率大，车就有更快的加速性能，而且可以保持较长的一段时间。 4、作为一辆车，Model S有哪些优点？这些优点是电动车带来的吗？ 作为一辆车，Model S主要具有以下几个优点 （1）起步加速快，顶配版本0-100公里加速4秒多，能战宝马M5

TESLA特斯拉解析

TESLA 硅谷工程师、资深车迷、创业家马丁·艾伯哈德（Martin Eberhard）在寻找创业项目时发现，美国很多停放丰田混合动力汽车普锐斯的私家车道上经常还会出现些超级跑车的身影。他认为，这些人不是为了省油才买普锐斯，普锐斯只是这群人表达对环境问题的方式。于是，他有了将跑车和新能源结合的想法，而客户群就是这群有环保意识的高收入人士和社会名流。 2003年7月1日，马丁·艾伯哈德与长期商业伙伴马克·塔彭宁（Marc Tarpenning）合伙成立特斯拉（TESLA）汽车公司，并将总部设在美国加州的硅谷地区。成立后，特斯拉开始寻找高效电动跑车所需投资和材料。

由于马丁·艾伯哈德毫无这方面的制造经验，最终找到AC Propulsion公司。当时，对AC Propulsion公司电动汽车技术产生兴趣的还有艾龙·穆思科（Elon Musk）。在AC Propulsion公司CEO汤姆·盖奇（Tom Gage）的引见下，穆思科认识了艾伯哈德的团队。2004年2月会面之后，穆思科向TESLA投资630万美元，但条件是出任公司董事长、拥有所有事务的最终决定权，而艾伯哈德作为创始人任TESLA的CEO。 在有了技术方案、启动资金后，TESLA开始开发高端电动汽车，他们选择英国莲花汽车的Elise作为开发的基础。没有别的原因，只是因为莲花是唯一一家把TESLA放在眼里的跑车生产商。

艾伯哈德和穆思科的共同点是对技术的热情。但是，作为投资人，穆思科拥有绝对的话语权，随着项目的不断推进，TESLA开始尝到“重技术研发轻生产规划、重性能提升轻成本控制”的苦果。2007年6月，离预定投产日期8月27日仅剩下两个月时，TESLA还没有向零部件供应商提供Roadster的技术规格，核心的部件变速箱更是没能研制出来。另一方面，TESLA在两个月前的融资中向投资人宣称制造Roadster的成本为6.5万美元，而此时成本分析报告明确指出Roadster最初50辆的平均成本将超过10万美元。 生意就是生意，尤其硅谷这样的世界级IT产业中心，每天都在发生一些令人意想不到的事情。投资人穆思科以公司创始人艾伯哈德产品开发进度拖延、成本超支为由撤销其

特斯拉电动汽车电池管理系统解析

1. Tesla目前推出了两款电动汽车，Roadster和Model S，目前我收集到的Roadster的资料较多，因此本回答重点分析的是Roadster的电池管理系统。 2. 电池管理系统（Battery Management System, BMS）的主要任务是保证电池组工作在安全区间内，提供车辆控制所需的必需信息，在出现异常时及时响应处理，并根据环境温度、电池状态及车辆需求等决定电池的充放电功率等。BMS的主要功能有电池参数监测、电池状态估计、在线故障诊断、充电控制、自动均衡、热管理等。我的主要研究方向是电池的热管理系统，因此本回答分析的是电池热管理系统 (Battery Thermal Management System, BTMS). 1. 热管理系统的重要性 电池的热相关问题是决定其使用性能、安全性、寿命及使用成本的关键因素。首先，锂离子电池的温度水平直接影响其使用中的能量与功率性能。温度较低时，电池的可用容量将迅速发生衰减，在过低温度下（如低于0°C）对电池进行充电，则可能引发瞬间的电压过充现象，造成内部析锂并进而引发短路。其次，锂离子电池的热相关问题直接影响电池的安全性。生产制造环节的缺陷或使用过程中的不当操作等可能造成电池局部过热，并进而引起连锁放热反应，最终造成冒烟、起火甚至爆炸等严重的热失控事件，威胁到车辆驾乘人员的生命安全。另外，锂离子电池的工作或存放温度影响其使用寿命。电池的适宜温度约在10~30°C 之间，过高或过低的温度都将引起电池寿命的较快衰减。动力电池的大型化使得其表面积与体积之比相对减小，电池内部热量不易散出，更可能出现内部温度不均、局部温升过高等问题，从而进一步加速电池衰减，缩短电池寿命，增加用户的总拥有成本。 电池热管理系统是应对电池的热相关问题，保证动力电池使用性能、安全性和寿命的关键技术之一。热管理系统的主要功能包括：1）在电池温度较高时进行有效散热，防止产生热失控事故；2）在电池温度较低时进行预热，提升电池温度，确保低温下的充电、放电性能和安全性；3）减小电池组内的温度差异，抑制局部热区的形成，防止高温位置处电池过快衰减，降低电池组整体寿命。 2. Tesla Roadster的电池热管理系统 Tesla Motors公司的Roadster纯电动汽车采用了液冷式电池热管理系统。车载电池组由6831节18650型锂离子电池组成，其中每69节并联为一组（brick），再将9组串联为一层（sheet），最后串联堆叠11层构成。电池热管理系统的冷却液为50%水与50%乙二醇混合物。 图 1.(a)是一层（sheet）内部的热管理系统。冷却管道曲折布置在电池间，冷却液在管道内部流动，带走电池产生的热量。图 1.(b)是冷却管道的结构示意图。冷却管道内部被分成四个孔道，如图 1.(c)所示。为了防止冷却液流动过程中温度逐渐升高，使末端散热能力不佳，热管理系统采用了双向流动的流场设计，冷却管道的两个端部既是进液口，也是出液口，如图 1(d)所示。电池之间及电池和管道间填充电绝缘但导热性能良好的材料（如Stycast 2850/ct），作用是：1)将电池与散热管道间的接触形式从线接触转变为面接触；2)有利于提高单体电池间的温度均一度；3)有利于提高电池包的整体热容，从而降低整体平均温度。

Tesla Model S电池组设计全面解析

Tesla Model S电池组设计全面解析 对Tesla来说最近可谓是祸不单行；连续发生了3起起火事故，市值狂跌40亿，刚刚又有3名工人受伤送医。Elon Musk就一直忙着到处“灭火”，时而还跟公开表不对Tesla“不感冒”的乔治·克鲁尼隔空喊话。在经历了首次盈利、电池更换技术·穿越美国、水陆两栖车等头条新闻后，Elon Musk最近总以各种负面消息重返头条。这位"钢铁侠。CE0在201 3年真是遭遇各种大起大落。 其中最为人关注的莫过于Model S的起火事故，而在起火事故中最核心的问题就是电池技术。可以说，牵动Tesla股价起起落落的核心元素就是其电池技术，这也是投资者最关心的问题。在美国发生的两起火事故有着相似的情节Model S 撞击到金属物体后，导致电池起火，但火势都被很好地控制在车头部分。在墨西哥的事故中，主要的燃烧体也是电池；而且在3起事故中，如何把着火的电池扑灭对消防员来说都是个难题。 这让很多人产生一个疑问：Model S的电池就这么不禁撞吗?在之前的一篇文章中，我跟大家简单讨论了一下这个问题，但只是停留在表面。读者普遍了解的是，Model S的电池位于车辆底部，采用的是松下提供的18650钴酸锂电池，整个电池组包含约8000块电池单元；钴酸锂电池能量密度大，但稳定性较差，为此Tesla研发了3级电源管理体系来确保电池组正常运作。现在，我们找到了Tesla的一份电池技术专利，借此来透彻地了解下Model S电池的结构设计和技术特征。 电池的布局与形体

FIG3 如专利图所示，Model S的电池组位于车辆的底盘，与轮距同宽，长度略短于轴距。电池组的实际物理尺寸是：长2.7m，宽1.5m，厚度为0.1 m至0.1 8m。其中0.1 8m较厚的部分是由于2个电池模块叠加而成。这个物理尺寸指的是电池组整体的大小，包括上下、左右、前后的包裹面板。这个电池组的结构是一个通用设计，除了18650电池外，其他符合条件的电池也可以安装。此外，电池组采用密封设计，与空气隔绝，大部分用料为铝或铝合金。可以说，电池不仅是一个能源中心，同时也是Model S底盘的一部分，其坚固的外壳能对车辆起到很好的支撑作用。 由于与轮距等宽，电池组的两侧分别与车辆两侧的车门槛板对接，用螺丝固定。电池组的横断面低于车门槛板。从正面看，相当于车门槛板"挂着。电池组。其连接部分如下图所示。 FIG, 4

特斯拉Model S电动汽车性能介绍

特斯拉Model S 特斯拉Model S并非小尺寸、动力不足的短程汽车——这是某些人对电动车的预期。作为特斯拉三款电动车中体积最大的车型，根据美国环保署认证，这款快捷、迷人的运动型轿车一次充电能够行驶265英里(426公里)，不过特斯拉声称可以达到300英里。不管哪种情况，这肯定是电动车行业的新高。Model S Performance版本的入门级价格为94,900美元，我测试的版本价格为101,600美元(按照美国联邦税收抵免，可以在此基础上扣减7,500美元)。 在一次开放驾驶上，这款特斯拉汽车硕大的85千瓦时电池的确可以至少行驶426公里。电流来自于车底的电池组，里面有大约7,000颗松下锂电池，重量约为590公斤(1,300磅). 试驾的第二天是前往威斯康辛州，在行驶了320公里后电几乎用光，不过其中包括了在芝加哥的一场交通拥堵中无奈爬行的两个小时。这天的测试充满野心，更多是针对性能而非行驶里程，包括这款特斯拉汽车迅速地用4.4秒时间从0加速到时速97公里(0至60英里每小时)，此外测试达到的最高时速为210公里。 我有没有提到，在0到时速100英里的加速时间方面，这款310千瓦(416马力)的特斯拉汽车将击败威力巨大、使用汽油的413千瓦(554马力)宝马M5？部分原因在于这款特斯拉汽车的同步交流电发动机能够即时提供600牛·米(443英尺磅)的扭矩。像电灯开关一样轻点特斯拉的油门，最大的扭矩已经准备就绪，一分钟内能够实现从0到5,100转。后悬挂、液冷式发动机可以保持1.6万转每分钟，通过一个单速变速箱将动力传导至后轮。 它就像一头冷酷的猛兽，在出奇安静之中让内燃机这个猎物消失于无形——安静到何种程度呢？来自轮胎和风阻的声音比在其他大部分豪华车中感受到的更加明显。安装于车底的电池让特斯拉获得与很多超级车相当的重心，这非常有利于稳定操控。Model S经过弯道的时候也能很好地保持贴地感。 尽管这款特斯拉汽车看起来并不笨重，但其重量达到2,108公斤；随着速度和重力的提升，这些多余的重量表露无遗。加大油门后，沉重的尾部会产生震动。在操控手感的愉悦性方面，特斯拉无法与宝马相提并论，甚至连马自达都赶不上。 美妙的试驾体验在你进入车内之前就开始了，你靠近汽车时，可伸缩的车门把手自动弹出。接着看到的是特斯拉标志性的驾驶室特色内容，一个43厘米(17英寸)电容触摸屏，看起来就像一对相互配合的iPad. 在其用铝合金加强的底盘和车身内，Model S可以容纳5人。一个可爱但是奇怪的按钮可以在车门位置增加脸朝车后的儿童座椅，从而实现最多承载7人。将第二排座椅向下折，可以扩展后座载货空间，可用于家得宝(Home Depot)采购之旅。由于引擎盖下面没有发动机，这些空间可以作为有用的前置行李箱，特斯拉将其称为“前备箱”(“frunk”)，就像保时捷911一样。

特斯拉纯电动车

目录 一、特斯拉简介 (3) 二、特斯拉纯电动车主要功能特点 (3) （一）Model S 主要特点 (3) （二）Model X 主要特点 (9) （三）Model 3 主要特点 (12) 三、特斯拉的电池技术 (13) （一）特斯拉动力电池简介 (13) （二）85kwh电池板的拆解分析 (14) （三）单体电池的能量密度 (20) （四）电量的衰减性能 (22) （五）电池检测实验室：从源头保证锂电池单体一致性 (24) （六）动力电池系统PACK技术 (25) （七）电池管理系统（BMS） (27) 四、特斯拉的充电技术 (35) （一）家用充电桩 (35) （二）超级充电桩 (37) （三）目的地充电桩 (38) （四）计划使用太阳能为超级充电站供电 (38) 五、电机及电控的主要技术 (38) （一）感应电机与永磁电机的对比 (39) （二）Model S采用三相交流感应电机 (40)

（三）双电机可以有效减少高速时的效率降低，并延长续航能力 (41) （四）电机的结构改进提效并易于自动化 (41) （五）逆变器采用分散塑封IGBT，实现低散热要求 (43) 六、车身的主要技术 (46) （一）全铝车身 (46) （二）Model X的双铰链鹰翼门 (47) 七、安全方面的主要技术 (48) （一）车身的安全设计 (49) （二）电池的安全性 (50) （三）信息安全技术 (51) 八、智能化技术 (51) （一）空中升级 (51) （二）远程诊断 (52) （三）自动求助 (52) （四）交互关系 (52)

特斯拉纯电动车的核心技术分析 一、特斯拉简介 特斯拉(Tesla)，是一家美国电动车及能源公司，产销电动车、太阳能板、及储能设备。总部位于美国加利福尼亚州硅谷帕洛阿尔托（Palo Alto）。 特斯拉第一款汽车产品Roadster发布于2008年，为一款两门运动型跑车。2012年，特斯拉发布了其第二款汽车产品——Model S，一款四门纯电动豪华轿跑车；第三款汽车产品为Model X，豪华纯电动SUV ，于2015年9月开始交付。特斯拉的下一款汽车为Model 3，首次公开于2016年3月，并将于2017年末开始交付。 2016年11月17日特斯拉电动车收购美国太阳能发电系统供应商SolarCity，使得特斯拉转型成为全球唯一垂直整合的能源公司，向客户提供包括Powerwall能源墙、太阳能屋顶等端到端的清洁能源产品。2017年2月1日，特斯拉汽车公司（Tesla Motors Inc．）正式改名为特斯拉（Tesla Inc．）。这意味着汽车不再是特斯拉的唯一业务。 二、特斯拉纯电动车主要功能特点 （一）Model S 主要特点 得益于特斯拉独特的纯电动动力总成，Model S 的性能表现十分出色，0-100公里/小时加速最快仅需2.7 秒。通过Autopilot 自动辅助驾驶（选装），Model S 还可以使高速公路驾驶更为安全且轻松，让你更好的享受驾驶乐趣。

深度揭秘特斯拉Model S底盘：电池组电机四驱

深度揭秘特斯拉Model S底盘：电池组/电机/四驱 特斯拉的第一代产品Roadster，用的是莲花Elise的底盘。这台车当时卖了2000多台。现在，这个经典的跑车底盘又被底特律电动车（Detroit Electric）拿来做另外一款“Roadster”了。 2012年，特斯拉发布Model S。底盘结构由特斯拉自主研发，并为其今后的车系奠定了基础。与燃油汽车不同，特斯拉一个底盘就可以涵盖所有级别的车型。比如将于2017年上市的Model 3，其底盘是在Model S的基础上缩短了轴距而已。 本期，我们来彻底解构下特斯拉Model S的底盘结构。共分为三部分来讲：电池组、电机，以及四驱。先从电池组说起。 特斯拉的电池，是特斯拉的核心专利技术之一，可以说是整台Model S最核心的一个零件。特斯拉一共拥有249项专利，其中有104项是跟电池有关的。与很多采用几个大的电池单元成电池组的布局不同，特斯拉采用的是与笔记本一样的电池。整台Model S的整备质量为2108kg（2.1吨），其中电池组的重量就占了600kg（0.6吨）。作为一辆D级豪华车，特斯拉Model S并没有超重。这在很大程度上得益于Model S的全铝车身。

由于电池组横贯于位于车辆底部，这使得Model S的重心得以降低，平衡了配重，从而提升了操控性。根据官方数据，Model S的前后配重比为48：52。 在Model S刚上市时，按照电池划分共有3款型号，分别是85kWh、60kWh，以及40kWh。2013年，由于40kWh车型销量惨淡，特斯拉决定停止销售。不久前，特斯拉又推出了70Kwh车型，来取代之前的60kWh版本。 值得一提的是，当年60kWh的车型与40kWh的车型，电池组其实是一样的；两者的区别在于，特斯拉将40kWh的电池进行了软件限制，从而在一个可容纳60kWh电量的电池组中，只有40kWh的电量可用。 而85kWh电池与60kWh电池的区别，主要是电池组中装配的电池单元数量。85kWh的电池组电压为400V，由一共16个电池包组成，每个电池包装配了444颗电池单元，所以这个电池组一共有7104颗电池组成。60kWh，则是由14个电池包，共计6216颗电池单元组成。这里所说的电池单元，是由松下提供的 NCR-18650A型电池。 18650是可充电锂离子电池的一种型号，它的命名来源于这种电池的尺寸 --18mm*65mm，但由于还要加入保护电路，所以电池的实际尺寸要略微大几零点几毫米。18650电池的主要用途，是笔记本电脑的电池，它有很多生产厂商；而特斯拉则选用了松下提供的18650电池，但要注意特斯拉使用的电池与笔记本中的电池还是有差别。18650只是一个统称。

特斯拉电动车2013全球销量

特斯拉电动车2013全球销量

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特斯拉的2013年：利润超1亿美元售车2.25万辆 2014年02月20日来源：第一电动网 特斯拉（NASDAQ:TSLA）将2013年一季度创出的盈利“奇迹”延续到了全年。按照特斯拉一贯采纳的非通用会计准则（Non-GAAP），特斯拉2013年赚得超过1亿美元的利润。特斯拉的电动汽车销量也大为增长，达到了约2.25万辆。 近日，特斯拉发布财务数据称，根据GAAP准则，即不计入股权奖励支出及其他一次性项目，特斯拉2013年营业收入为20.13496亿美元，对比2012年的4.13256亿美元，同比增长387.2%。而按照非GAAP准则，特斯拉2013年营业收入为24.77662亿美元，对比2012年的4.13256亿美元，同比增长499.5%。 根据GAAP准则，特斯拉去年亏损额为7401.4万美元，2012年则为39621.3万美元，同比削减81.3%。按照非GAAP准则，特斯拉去年实现利润10356.3万美元，2012年亏损34421.4万美元。 特斯拉汽车于美国时间本周三下午发布了2013年的致股东邮件。邮件显示，第四季度，特斯拉创纪录地销售了6892辆电动汽车，全年销量22477辆。 未来，特斯拉还计划在美国发展超级充电站网络和服务中心，推动汽车销售。此外特斯拉还预计，欧洲和中国市场将带来巨大销量。2014年的汽车总销量将达到3.5万辆，比今年的22477辆高55%。

特斯拉分析报告

特斯拉分析报告 Revised as of 23 November 2020

目录 特斯拉电动汽车国际发展分析报告 综合经营教育 组织：市场策划1301 班 指导老师：胡子娟 组长：符美丹 组员：徐宝怡、李嘉尊、张家梦、杨伟怡 华南农业大学珠江学院 电话： 乐享科技 2016-4-6

一、背景 (一)公司概况 2003年7月1日，马丁艾伯哈德与长期商业伙伴马克塔彭宁合伙成立特斯拉（TESLA）汽车公司，并将总部设在美国加州的硅谷地区2004年2月，埃隆马斯克向特斯拉投资630万美元，但条件是出任公司董事长、拥有所有事务的最终决定权，而马丁艾伯哈德作为特斯拉之父任公司的CEO。不可忽视的是，特斯拉的背后，站着众多超级投资人。其中包括谷歌创始人拉里佩奇、谢尔盖布林等人，还包括丰田、戴姆勒奔驰的子公司和松下等传统汽车巨头。松下是特斯拉的锂电池电芯供应商，而特斯拉汽车的部分设计也受益于奔驰的启发特斯拉刷新了世界对电动汽车的认知，从这一点出发，特斯拉可以称得上是一个改变了世界的公司。特斯拉当前的创新应该更多在商业模式以及对电动汽车的发展的推动上，是一个令人充满期待，并且值得让人敬佩的公司。从诞生之日起，特斯拉的品牌一直都与“环保”、“高科技”等标签贴在一起，时时闪现出高冷的明星气质。这的确在品牌初期为其吸引了众多支持者，并获得了意想不到的营销效果。而借助这层光环加持，特斯拉开始了自己的故事。在本土市场较为稳定之后特斯拉开始开拓中国市场。 (二)公司产品 1.T esla Roadster 2.T esla Model S 3.T esla Model X

(完整版)特斯拉汽车案例介绍

特斯拉汽车案例介绍 一、 1、发展背景 2003年在美国硅谷成立了一家汽车公司，这个在选址上独具一格的传统汽车公司名为特斯拉，在企业一开始发展的阶段就将公司的选址放在美国西部的科技圣地——硅谷，这个二十一世纪电子和计算机业的王国，突然诞生了一家汽车公司，于周围的企业显得格格不入，但就在这样的环境下，从硅谷走出了一辆通向未来的汽车。 在众多传统巨头坚持不住的时候，特斯拉默默无闻的坚持了下来，并且发展的如火如荼，目前特斯拉的股票突破了100美元大关，直追日本丰田，成为了美国股市之中为数不多的超过100美元的汽车公司，超越了众多的汽车行业巨头。这个不太出名的小汽车企业是如何发展起来的呢？ 在1990年，由美国通用汽车研发并制造了第一款现代化电动汽车EV-1，这款低风阻、双门双座的电动汽车却采用租赁的方式对外进行，大多数租户第一次接触到现代电动汽车，对EV-1表现得尤为满意，但EV-1的结局却让人感慨万千，由于这款车的投入和产量不大，在生产一千多台后停止生产，1999年通用回收销毁EV-1，让租户们很不理解，大多数租户都愿意对租的车进行购买，最终全部被通用回收，分批销毁，最后只有几台放置在博物馆。参于EV-1的工程师不甘心失败，于是创建研究铅酸电池的AC Propulsion汽车公司，由于研发铅酸电池一直没大规模突破，马丁·艾伯哈德投资了15万美元，他希望尝试用笔记本电脑的锂电池作为电动汽车电池， 艾伯哈德劝说AC Propulsion公司为他制造一辆电动汽车，就这样科尼在无意成立汽车公司。艾伯哈德于是决定自己来。艾伯哈德在寻找创业项目时发现，停放跑车的私家车道上常有着丰田混合动力汽车的身影。艾伯哈德觉得，所以，有了将跑车和电动汽车结合的主意。2003年7月1日，马丁·艾伯哈德于长期商业伙伴马克·塔彭宁合伙成立特斯拉（TESLA）汽车公司，并将总部设在美国加州的硅谷地区。 2公司现状 特斯拉企旗下现售四款电动汽车，以经营高性能纯电动汽车为主，早在2016年年营业额突破了70亿，说起电动汽车，在这个领域内特斯拉却是行业中的大头，处于翘楚地位，无人驾驶技术较为先进成熟，量产出L3级高度驾驶系统，同样是搭载锂电池的特斯拉续航能力远远超越其他同类电动汽车，是目前新能源汽车领域的佼佼者，更是当前新能源行业的领头羊。在2018全球电动汽车销量排行中特斯拉汽车占据了前五位中的三个，市场份额就占据了整个市场中的11%，可以彰显出特斯拉在电动汽车行业的领导地位。

Tesla motor特斯拉电动汽车分析

Tesla Motors Norbert Binkiewicz Justin Chen Matt Czubakowski June 4, 2008

1SWOT Analysis Strengths ?Good engineering and technology research capability ?Able to raise large amounts of capital ?First mover advantage; the first company to offer a relatively practical fully electric car, customers include high-profile figures like Arnold Schwarzenegger, George Clooney, and Jay Leno ?Designs and builds many of the components in its cars, including the power electronics, motor and battery packs Weaknesses ?Doesn’t have much brand recognition among the general public ? A very small company with small sales volume, so no economies of scale ?Possible supply problems with components, especially if demand increases ?The Tesla Roadster hasn’t been on the market for very long, the longevity of fully electric cars remains to be proven Opportunities ?Moving towards the family sedan market and making a product that is meant for more of the automotive market ?Price of oil and gasoline skyrocketing, making the price premium for an electric car less of an issue ?Expanding into developing lithium-ion batteries and other energy technologies, partnering with a battery company to improve battery technology Threats ?Wrightspeed X1, a prototype high performance electric car that caters to the same market; the only direct competitor to Tesla that offers a similar product ?Large automobile companies entering the market with full and hybrid electric cars, the GM Volt and Toyota Prius ?The price of oil falling dramatically in the short run ? A competitor having a breakthrough in related energy technologies, like hydrogen powered cars, natural gas, or ethanol

特斯拉家用充电桩参数及规格

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